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Basic Description

In 1899, C. N. Little published a table of 43 nonalternating knots of 10 crossings that listed these two knots as being distinct. Seventy-five years later, Kenneth Perko, a lawyer and part-time mathematician, discovered that these were actually the same knot[1].

To say that two knots are the same is to say that one can be deformed into the other without breaking the knot or passing it through itself. To prove that two knots are the same, we can create one of them out of actual rope, and tug at it and move it around until it looks like the other. As the story goes, that's how Perko figured out that these knots are the same - by working with rope on his floor.

We can also prove that two knots are the same by working with their projections. A projection of a knot is a flat representation of it, essentially a 2D drawing of the knot. There are many ways to use projections to show that certain knots are distinct from each other, but the main way of using projections to demonstrate that two knots are the same is to use the Reidemeister moves, which are described below.

A More Mathematical Explanation

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Reidemeister moves

As was stated above, knots are considered to be the same if one can be rearra [...]

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Reidemeister moves

As was stated above, knots are considered to be the same if one can be rearranged into the other without breaking the string or passing it through itself. This kind of transformation is called an ambient isotopy. So if we want to prove that two knots are the same, we need to prove that we can get from one to the other by means of an ambient isotopy. One way to do this is to tie the knots out of rope or extension cords and physically re-arrange them. But what if we want a written proof, one that we can look at and share with other people? We need some way to prove that knots are the same without having to physically create them.

To do this we will need to work with a knot’s projection. The projection of a knot is two-dimensional image of it, showing what the knot would look like if it was projected onto a plane. One knot can have multiple projections, because we can create a projection while looking at the knot from any point of view, or after moving the knot trough any ambient isotopy.

Now that we know what projections and ambient isotopies are, our question is: what manipulations can we make on a knot’s projection that correspond to ambient isotopies in three dimensions?

The first answer is a planar isotopy. A planar isotopy is the sort of transformation you could make if the projection of a knot was printed on very stretchy rubber. The image can be stretched in all directions, but none of the crossings are affected:

The original image.

These are planar isotopies.

This is not a planar isotopy.

The second answer is the Reidemeister moves, a set of three changes you can make to a knot’s projection that do affect the knot’s crossings but are still ambient isotopies:

Type I Reidemeister Move:

The first Reidemeister move allows you to create a twist in a strand that goes in either direction.

Type II Reidemeister move:

The second Reidemeister move allows you to slide one strand on top of or behind another.

Type III Reidemeister move:

The third Reidemeister move allows you to slide a strand to the other side of a crossing.

Proving that the Perko pair are equivalent

In his paper "On the Classification of Knots", Kenneth Perko provided an abridge proof that the knots now known as Perko A and Perko B are the same. This proof is shown below:

Perko's proof relies on the ability of the reader to manipulate the knots in their head and verify that each projection can be manipulated to look like the next. To create a full, rigorous proof, we need to use planar isotopies and the Reidmeister images, as described above.

Below is a step-by-step Reidemeister moves proof that follows the outline of Perko's shorter proof. The arrows between each step are labeled to show how we get from one image to the other: "p.i." means we use a planar isotopy, "I" means we use the first Reidemeister move, "II" means we use the second move, and "III" means we use the third move. Mousing over a step will highlight the part of the knot that's about to move in pink, and display a dotted green line showing where it will move to.

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Why It's Interesting

Interest in comparing and tabulating knots actually grew out of a mistaken theory in chemistry. In the 1880s, chemists believed that space was filled with something called ether, and Lord Kelvin proposed that different atoms were different types of knots in the ether[1].

Belief in ether soon faded, but interest in identifying and cataloging different types of knots stayed. In 1899, the Perko knots were published as distinct knots in C. N. Little's table of 43 nonalternating knots of 10 crossings[1]. These knots were believed to be distinct until 1974, when Kenneth Perko published a paper titled "On the Classification of Knots" that demonstrated that they were in fact the same.

The Perko knots are sometimes used to illustrate the point that knot theory is still a relatively young mathematical field, and that it is accessible to non-mathematicians.